What sorting method is the easiest to implement? and which is most effiecent?

That's like asking what the fastest vehicle is. Anyone can quote high-powered machines back at you but we don't know if you're asking about acceleration or top speed, whether on-road or off-road...? Motorbike, jetboat, car, plane or even space shuttle...?

There are a few easy ones to implement, but it depends on the data structure. For example Bubble Sort is easy for an array, but it is quite tricky for a linked list. Some algorithms are easy for one data structure but are basically impossible for other data structures.
Some algorithms may not suit because they don't respect the existing ordering of items that have the same key value, i.e algorithms that are not stable. E.g. Quicksort is typically fast, but is not stable.
Some algorithms can be efficient time-wise, and reasonably easy to implement, but take too much RAM.
Even the shape of the distribution of the values, the number of duplicates, and the initial ordering (number of inversions) play an important part in how long the sorting takes.

If you want to overall easiest to implement I'd say Insertion Sort (works on more data structures). If you want the fastest algorithm for a huge number of items it's probably Pigeonhole Sort. If you want the fastest algorithm for a small number of items then its definitely something else entirely. In practice you usually shouldn't or often even cant, use either of those for what you're sorting, depending on the data type or item count.

I fully realise that you couldn't know what to ask when you don't know any of the above about sorting. However you've now been made aware that there is no real answer to the general question about sorting speed. Here's some of what we need to know to give you the best answer:

What type of data structure - array/list/doubly-linked-list etc?

How many items could there be?

Will they usually be mostly sorted already?

What is the data type?

Could there be duplicates?

Okay you did mention you're sorting a vector later on so that's the first question answered. In case you cant wait for another reply though, Shell Sort and Comb Sort are fairly easy to implement, perform quite well for any number of items, and take no extra memory at all.